136 citations · 451 across the 26 of their papers we have counts for
30 papers
Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the -sense
Julia Ackermann, Arnulf Jentzen, Thomas Kruse +2
Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have gener…
The Collective Dynamics of a Stochastic Port-Hamiltonian Self-Driven Agent Model in One Dimension
Matthias Ehrhardt, Thomas Kruse, Antoine Tordeux
The collective motion of self-driven agents is a phenomenon of great interest in interacting particle systems. In this paper, we develop and analyze a model of agent motion in one…
Nonlinear Monte Carlo methods with polynomial runtime for Bellman equations of discrete time high-dimensional stochastic optimal control problems
Christian Beck, Arnulf Jentzen, Konrad Kleinberg +1
Discrete time stochastic optimal control problems and Markov decision processes (MDPs), respectively, serve as fundamental models for problems that involve sequential decision maki…
Reducing Obizhaeva-Wang type trade execution problems to LQ stochastic control problems
Julia Ackermann, Thomas Kruse, Mikhail Urusov
We start with a stochastic control problem where the control process is of finite variation (possibly with jumps) and acts as integrator both in the state dynamics and in the targe…
Self-exciting price impact via negative resilience in stochastic order books
Julia Ackermann, Thomas Kruse, Mikhail Urusov
Most of the existing literature on optimal trade execution in limit order book models assumes that resilience is positive. But negative resilience also has a natural interpretation…
Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse +1
Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applicati…
On the speed of convergence of Picard iterations of backward stochastic differential equations
Martin Hutzenthaler, Thomas Kruse, Tuan Anh Nguyen
It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearity conv…
Multilevel Picard approximations for McKean-Vlasov stochastic differential equations
Martin Hutzenthaler, Thomas Kruse, Tuan Anh Nguyen
In the literatur there exist approximation methods for McKean-Vlasov stochastic differential equations which have a computational effort of order . In this article we introduce…
Inhomogeneous affine Volterra processes
Julia Ackermann, Thomas Kruse, Ludger Overbeck
We extend recent results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an…
Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse +1
The recently introduced full-history recursive multilevel Picard (MLP) approximation methods have turned out to be quite successful in the numerical approximation of solutions of h…
Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations
Christian Beck, Arnulf Jentzen, Thomas Kruse
The approximative calculation of iterated nested expectations is a recurring challenging problem in applications. Nested expectations appear, for example, in the numerical approxim…
Càdlàg semimartingale strategies for optimal trade execution in stochastic order book models
Julia Ackermann, Thomas Kruse, Mikhail Urusov
We analyze an optimal trade execution problem in a financial market with stochastic liquidity. To this end we set up a limit order book model in continuous time. Both order book de…
Optimal trade execution in an order book model with stochastic liquidity parameters
Julia Ackermann, Thomas Kruse, Mikhail Urusov
We analyze an optimal trade execution problem in a financial market with stochastic liquidity. To this end we set up a limit order book model in which both order book depth and res…
Properties of the EMCEL scheme for approximating irregular diffusions
Stefan Ankirchner, Thomas Kruse, Wolfgang Löhr +1
We prove several properties of the EMCEL scheme, which is capable of approximating one-dimensional continuous strong Markov processes in distribution on the path space (the scheme…
Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
Partial differential equations (PDEs) are a fundamental tool in the modeling of many real world phenomena. In a number of such real world phenomena the PDEs under consideration con…
Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
Christian Beck, Fabian Hornung, Martin Hutzenthaler +2
One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministi…
Approximating exit times of continuous Markov processes
Thomas Kruse, Mikhail Urusov
The time at which a one-dimensional continuous strong Markov process attains a boundary point of its state space is a discontinuous path functional and it is, therefore, unclear wh…
Wasserstein convergence rates for random bit approximations of continuous Markov processes
Stefan Ankirchner, Thomas Kruse, Mikhail Urusov
We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of coin tossi…
A functional limit theorem for coin tossing Markov chains
Stefan Ankirchner, Thomas Kruse, Mikhail Urusov
We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability , respectively. The theorem e…
A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse +1
Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential…
Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse +2
For a long time it is well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort…
Multi-level Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities
Martin Hutzenthaler, Thomas Kruse
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) have a wide range of applications. In particular, high-dimensional PDEs with…
On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations
Weinan E, Martin Hutzenthaler, Arnulf Jentzen +1
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineerin…
Multilevel Picard iterations for solving smooth semilinear parabolic heat equations
Weinan E, Martin Hutzenthaler, Arnulf Jentzen +1
We introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time p…